How do you derive f = D² / (16 d)?

Place the vertex at the origin so the parabola is y = x² / (4f). A rim point of the reflector is at (D/2, d). Substituting gives d = (D/2)² / (4f) = D² / (16f), and solving for f gives f = D² / (16 d).

Which units should I use in the calculator?

Use the same unit for both the depth d and the diameter D — for example, both in centimeters, both in meters, or both in inches. The focal distance f is then reported in that same unit.

Focus of a Parabolic Reflector Calculator

Q: What is the formula for the focal distance of a parabolic reflector?

f = D² / (16 d), where D is the diameter of the reflector and d is its depth. The focal distance f is measured from the vertex to the focus, in the same unit as D and d.

Q: What is the difference between focal distance and focal length?

In this context they are the same quantity: the distance from the vertex of the parabola to its focus. 'Focal length' is the more common term in antenna engineering; 'focal distance' is more common in geometry.

Formula for the Focal Distance

A parabola with a vertical axis and its vertex at the origin has the equation:

\[ y = \frac{1}{4f} x^2 \]

where \( f \) is the focal distance — the distance between the vertex \( V \) and the focus \( F \).

For a parabolic reflector with diameter \( D \) and depth \( d \), placing the vertex at the origin puts the outer rim at \( \left(\frac{D}{2}, d\right) \). Substituting:

\[ d = \frac{1}{4f} \left(\frac{D}{2}\right)^2 = \frac{D^2}{16f} \]

Solving for \( f \) gives the formula for the focal distance:

\[ f = \frac{D^2}{16 d} \]

Cross-section of a parabolic reflector showing the depth d, the diameter D, the focus F, and the focal distance f from vertex to focus — **Figure 1.** Cross-section of a parabolic reflector.

Calculate the Focal Distance

Enter the depth \( d \) and the diameter \( D \) as positive real numbers, then click Calculate. Use the same unit for both — the result is reported in that same unit.

Worked Example

Suppose a reflector has depth \( d = 35 \) cm and diameter \( D = 230 \) cm. Then:

\[ f = \frac{230^2}{16 \times 35} = \frac{52{,}900}{560} \approx 94.46 \text{ cm} \]

So the focus is about 94.46 cm from the vertex, along the axis of symmetry.

Frequently Asked Questions

What is the formula for the focal distance of a parabolic reflector?

\( f = \dfrac{D^2}{16d} \), where \( D \) is the diameter and \( d \) is the depth. The focal distance \( f \) is measured from the vertex to the focus, in the same unit as the inputs.

What is the difference between focal distance and focal length?

In this context, they are the same quantity — the distance from the vertex of the parabola to its focus. "Focal length" is more common in antenna engineering; "focal distance" is more common in geometry.

How do you derive \( f = \dfrac{D^2}{16d} \)?

Place the vertex at the origin so the parabola is \( y = x^2/(4f) \). A rim point is at \( (D/2, d) \). Substituting gives \( d = \dfrac{(D/2)^2}{4f} = \dfrac{D^2}{16f} \), and solving for \( f \) gives \( f = \dfrac{D^2}{16d} \).

Which units should I use?

Use the same unit for both the depth and the diameter — both in centimeters, both in meters, or both in inches. The focal distance is then reported in that same unit.

More Tools & Resources

Last updated: October 17, 2026